Is a relationship and sex between a Muslim woman and non Muslim man permitted in Islam?
Short answer: NO!
- First of all, sex before marriage constitute zina, between anyone - muslims or not.
وَمَن لَّمْ يَسْتَطِعْ مِنكُمْ طَوْلًا أَن يَنكِحَ الْمُحْصَنَاتِ الْمُؤْمِنَاتِ فَمِن مَّا مَلَكَتْ أَيْمَانُكُم مِّن فَتَيَاتِكُمُ الْمُؤْمِنَاتِ ۚ وَاللَّهُ أَعْلَمُ بِإِيمَانِكُم ۚ بَعْضُكُم مِّن بَعْضٍ ۚ فَانكِحُوهُنَّ بِإِذْنِ أَهْلِهِنَّ وَآتُوهُنَّ أُجُورَهُنَّ بِالْمَعْرُوفِ مُحْصَنَاتٍ غَيْرَ مُسَافِحَاتٍ وَلَا مُتَّخِذَاتِ أَخْدَانٍ ۚ فَإِذَا أُحْصِنَّ فَإِنْ أَتَيْنَ بِفَاحِشَةٍ فَعَلَيْهِنَّ نِصْفُ مَا عَلَى الْمُحْصَنَاتِ مِنَ الْعَذَابِ ۚ ذَٰلِكَ لِمَنْ خَشِيَ الْعَنَتَ مِنكُمْ ۚ وَأَن تَصْبِرُوا خَيْرٌ لَّكُمْ ۗ وَاللَّهُ غَفُورٌ رَّحِيمٌ
And whoever among you cannot [find] the means to marry free, believing women, then [he may marry] from those whom your right hands possess of believing slave girls. And Allah is most knowing about your faith. You [believers] are of one another. So marry them with the permission of their people and give them their due compensation according to what is acceptable. [They should be] chaste, neither [of] those who commit unlawful intercourse randomly nor those who take [secret] lovers. But once they are sheltered in marriage, if they should commit adultery, then for them is half the punishment for free [unmarried] women. This [allowance] is for him among you who fears sin, but to be patient is better for you. And Allah is Forgiving and Merciful.
وَلَا تَقْرَبُوا الزِّنَا ۖ إِنَّهُ كَانَ فَاحِشَةً وَسَاءَ سَبِيلًا
And do not approach unlawful sexual intercourse. Indeed, it is ever an immorality and is evil as a way.
and many other verses regarding this. So sex is only allowed under marriage.
- Second of all, regarding marraige. Only a disbeliever marries a disbeliever and an adulterer marries an adulterer:
الزَّانِي لَا يَنكِحُ إِلَّا زَانِيَةً أَوْ مُشْرِكَةً وَالزَّانِيَةُ لَا يَنكِحُهَا إِلَّا زَانٍ أَوْ مُشْرِكٌ ۚ وَحُرِّمَ ذَٰلِكَ عَلَى الْمُؤْمِنِينَ
The fornicator does not marry except a [female] fornicator or polytheist, and none marries her except a fornicator or a polytheist, and that has been made unlawful to the believers.
Basically Muslim are not permitted to marry a polytheist nor an adulterer, otherwise he/she will be considered as the same.
God made this world to test us. The real life is the next one. Being a test, you are given a free will to choose whatever you want. Whatever you choose, will have consequences in this world and the next.
As for sexual relation before marriage, please consider that fact that human being is one the weakest creature at his birth. He need the love of mother and care of father for a long time from his birth. This can only be provided in a family. Thus the institution of family is one of the basic needs for the survival of human race. Allah recognizes this fundamental need and thus given us instruction to uphold the sanctity of this institution. One of it that sexual relations are only allowed within marriage.
As for marrying a non-muslim, please consider that within a family, the man has the upper hand. it is observable across cultural and ethnic boundaries. Thus you will be putting yourself as well as your children in a difficult position, if he tries to force you out of it. You can say that he is not like that but, believe me, things CHANGE after marriage.
If he choose to embrace Islam, before marriage, then, hopefully, you (and your children) would be in a better position with regard to religion.